OFFSET
0,2
COMMENTS
When a prospective a(n) = k has prime factorization k = Product p(i)^r(i), its number of divisors is tau(k) = A000005(k)= Product (r(i)+1) and here this is to be tau(k) = 10^n.
Assigning exponents r(i) to primes p(i) in decreasing order minimizes k, but various ways to split 10^n into a product (r(i)+1) results in k of various size.
LINKS
EXAMPLE
a(1) = 48 = 2^4 * 3, and 48 is the least integer which has exactly 10^1 divisors(1,2,3,4,6,8,12,16,24,48).
MAPLE
# Construct list of ordered lists of factorizations of n with
# minimum divisors mind.
# Returns a list with A001055(n) entries if called with mind=2.
# Example: print(ofact(10^3, 2))
ofact := proc(n, mind)
local fcts, d, rec, r ;
fcts := [] ;
for d in numtheory[divisors](n) do
if d >= mind then
if d = n then
fcts := [op(fcts), [n]] ;
else
# recursive call supposed one more factor given
rec := procname(n/d, max(d, mind)) ;
for r in rec do
fcts := [op(fcts), [d, op(r)]] ;
end do:
end if;
end if;
end do:
return fcts ;
end proc:
A005179 := proc(n)
local Lexp, a, eList, cand, maxxrt ;
if n = 1 then
return 1;
end if;
Lexp := ofact(n, 2) ;
a := 0 ;
for eList in Lexp do
maxxrt := ListTools[Reverse](eList) ;
try
cand := mul( ithprime(i)^ ( op(i, maxxrt)-1), i=1..nops(maxxrt)) ;
catch:
# print("catch cand") ;
cand := 0 ;
end try ;
if a =0 or ( cand < a and cand > 0) then
a := cand ;
end if;
end do:
a ;
end proc:
A370515 := proc(n)
A005179(10^n) ;
end proc:
seq(A370515(n), n=0..10) ; # R. J. Mathar, Jun 06 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhining Yang, Feb 21 2024
STATUS
approved